International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
  
^ 
various portions of the test trajectory. Figure 3 shows the 
positional error of the integrated MEMS IMU and GPS during 
the simulated GPS outages. The figure clearly indicates that the 
IMU stand-alone results during the GPS outages are within 5-10 
m (RMS). This accuracy meets the general requirements for car 
navigation. A possible explanation for this surprising 
performance is the bias calibration procedure applied to the 
IMU. It successfully eliminated the long-term gyro bias. Since 
the short-term bias is much smaller, of the order of 50 deg/h, 
short outages like the ones simulated here can be bridged quite 
well. These results are confirmed by those in Hanse (2004) and 
Geen (2004). 
  
  
  
  
  
Figure 2: University of Calgary MEMS-based IMU (without 
case) 
i { North 
! | East 
i | Height 
GPS gap 
  
Position Error (m) 
-40 t 1 
-80 = i i i 1 i E 
162800 163000 163200 163400 163600 163800 164000 164200 164400 164600 164800 
GPS Time (sec) 
Figure 3: Positional Drift during GPS blockage 
4. MATHEMATICAL MODELING 
The formulation of the direct geo-referencing formula is rather 
straight-forward, for details see for instance Schwarz (2000). 
The standard implementation of this formula will, however, 
cause difficulties when low-accuracy gyros are used. The 
modifications necessary in this case will be discussed in this 
chapter. Figure 4 depicts airborne mobile mapping using a 
digital frame camera. The mathematical model is given in 
equation (1) and will be used as the standard model in the 
following discussion. The terms in the equation are listed in 
Table 1. 
m m. m p e e GPS 
EAT AP CES Ry (is, oR or + 4 INS — ans ] [1] 
GPS GPS antenna 
  
   
    
m 
Ry (t 
INS b-frame + 
c-frame 
  
  
m m m 
r (t)/ GPS La 
INS aan m mn dl 
f Or R, () = R, (0 Re 
Origin of ll 
v Camera Calibration 
nie attitude INS attitude 
Figure 4: Principle of Airborne Geo-referencing 
  
  
  
  
  
Variable Obtained from 
m is the coordinate vector of point (i) in the 
5 mapping frame (m-frame) Unknown (3) 
pP. s) is the interpolated coordinate vector of the 
navigation sensors (INS/GPS) in the m-frame 
i is a scale factor, determined by stereo 
5 techniques, laser scanners or DTM 
m is the interpolated rotation matrix between the 
Rp (1) navigation sensor body frame (b-frame) and 
the m-frame 
(t) is the time of exposure, ie. the time of 
  
  
  
  
  
  
  
  
capturing the images, determined by 
synchronization 
b is the differential rotation between the C-frame 
R. and the b-frame, determined by calibration 
pt is the coordinate vector of the point in the C- 
frame (i.e. image coordinate), 
pt vector between IMU center and camera 
INS principal point, determined by calibration 
, GPs vector between IMU center and GPS antenna 
INS center, determined by calibration 
  
Table 1: Elements of the Georeferencing Formula 
Implementation of this formula requires inertial and GPS 
measurements for the determination of the two time-dependent 
terms on the right-hand side of equation (1), as well as image 
coordinate measurements, in the c-frame, for the determination 
of the object point coordinate vector r^. The c-frame has its 
origin in the perspective centre of the camera, its z-axis is 
defined by the vector between the perspective centre and the 
principal point of the photograph, and its (x,y)-axes are defined 
in the plane of the photograph and are measured with respect to 
the principal point. The corresponding image vector is therefore 
of the form: 
XX 
c g h 
r-= y=, | where 
are the principal point coordinates 
(Xp, Yo) 
f is the camera focal length